Betti tables have been studied since early works on multiparameter persistence [1]}, [2]} where it was noted that there is an equivalence between the category of \(n\) -parameter persistence modules and the category of \(n\) -graded modules over the polynomial ring \(S:=\mathbb {F}[x_1, \ldots ,x_n]\) .
Explicitly, the correspondence takes a persistence module \(\lbrace  V^u ,\varphi ^{u,v} \rbrace \)  to the \(n\) -graded \(S\) -module \(\bigoplus _{u\in \mathbb {Z}^n} V^u\)  with the action of \(S\)  defined by \(x_i \cdot z = \varphi ^{u,u+e_i} (z)\) , for all \(z\in V^u\)  and all \(i\in [n]\) . This correspondence allows for the use of tools from commutative algebra to study persistence modules. We refer the reader to [3]} for background on such invariants for multigraded modules, while here we adopt the point of view of persistence modules.
